• Process of choosing between N alternatives \(\rightarrow\) N racing evidence accumulators

• Assumptions:

  • One threshold level of evidence
  • Random uniform distribution of starting evidence
  • Drift rate for each alternative
  • Non-decision time

• Lexical decision task \(\rightarrow\) word vs. non-word

• Participants: 4

• Trials: 10,000

• Conditions: Speed vs.Accuracy (between subjects design)

• DR implementation:

  • 250 ms to give second response
  • Participants were not instructed to give DRs (implicit)

• Additional information beyond response choice and RT

• Better understanding of the decision-making process as a whole

• Determine if including DR will make the model learn more

  • RQ: Does including DR improve posterior contraction

• MCMC more computationally costly

• BayesFlow: training is costly, but inference is fast

RDM (base model for all)

\[dx_i = [v_i]dt + [\sigma_i \epsilon]\sqrt{dt}\ \]

Feed-forward inhibition (FFI)

• Future evidence accumulation is reduced based on the accumulation rate of the competing alternative

Leaky-competing accumulator (LCA)

• Lateral inhibition with leakage + evidence cannot be negative
• Leakage: rate at which the alternative’s accumulated evidence is reduced

@nb.jit(nopython=True, cache=True)
def trial(drift, starting_point, boundary, ndt, max_t, max_drt=0.25, s=1, dt=None):
    drift = np.asarray(drift, dtype=np.float64)  # Convert before passing to JIT
    response = -1
    rt = -1

    if dt is None:
        dt = max_t / 10_000.0  # Ensure float division

    t = 0
    start = float(starting_point)  # Ensure float type
    evidence = np.random.uniform(0, start, size=len(drift))

    boundary += start  # Adjust boundary based on start
    dr = False  # Initialize double response

    # Initial evidence accumulation
    while np.all(evidence < boundary) and t < max_t:
        for resp in range(len(drift)):  # Normal loop (prange if parallel)
            evidence[resp] += dt * drift[resp] + np.random.normal(0, np.sqrt(s**2 * dt))
        t += dt

    rt = t + ndt
    drt = 0
    response_arr = np.where(evidence > boundary)[0]  # Avoid tuple issue

    if response_arr.size > 0:
        response = response_arr[0]  # Take first element
    else:
        response = -1  # Default

    # Double response accumulation
    while drt < max_drt and not dr:
        for resp in range(len(drift)):
            if response != -1 and resp != response:
                evidence[resp] += dt * drift[resp] + np.random.normal(0, s) * np.sqrt(dt)
                if evidence[resp] >= boundary:
                    dr = True
                    break
        drt += dt  # Only increase while dr is False

    return rt, response, dr, drt

Adding double response did kind of lead to the model to learn more

RDM

FFI

LCA

It also kind of improved the posterior contraction

RDM

FFI

LCA

• ⁠Priors can be more grounded in theory

• ⁠More participants, more double response

Accuracy Participant 2

• Study design \(\rightarrow\) explicit double responses

• Alternative definition of double response \(\rightarrow\) loser drift over takes winner drift